add the checkbox to your search form and set it to pass the activity ID as the value.
on the results page, go to bindings, click the plus button and select Form Data, then select the search form to add the search bindings to the results page.
in the data assist saerch behavior, set the Activity Criteria to use the Key Word search type. Set the OR seperator to use a coma followed by a space ", "
for the search value, click the lightning bolt icon and select the activity checkout box
Now, on the search form, edit the name of the activity check box to add "" at the end. This will create an array of the activity selections.
on the results page, add the following code to convert the array to a comma separated list:
<?php if(isset($_POST['activities'])) $_POST['activities'] = implode(", ", $_POST['activities']); ?>
the example above assumes the activities checkbox is named activities, if you use a different name, adjust accordingly.